A multivariate rational interpolation with no poles in R m

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1 NTMSCI 3, No., 9-8 (05) 9 New Treds i Mathematical Scieces A multivariate ratioal iterpolatio with o poles i R m Osma Rasit Isik, Zekeriya Guey ad Mehmet Sezer Departmet of Mathematics, Faculty of Educatio, Mugla Uiversity, Mugla, Turkey Departmet of Mathematics, Faculty of Scieces ad Arts,Maisa Celal Bayar Uiversity, Maisa, Turkey Received: 5 July 04, Revised: August 04, Accepted: 0 August 04 Published olie: December 04 Abstract: The aim of this paper is to costruct a family of ratioal iterpolats that have o poles i R m. This method is a extesio of Floater ad Hormas method []. A priori error estimate for the method is give uder some regularity coditios. Keywords: Ratioal iterpolat, Floater ad Hormas method, Error estimate. Itroductio Give a fuctio f defied o a m dimesioal box, we ca approximate f by polyomial iterpolatio. If the set of approximatig fuctios is exteded to the set of all ratioal fuctios, amely fuctios of the form p, p ad q q are ay polyomials, it is hard to cotrol the occurrece of poles ad to specify accuracy of the approximate solutio. Polyomial iterpolatio is thus a special case of ratioal iterpolatio. Hece, it ca be expected that ratioal iterpolatio may give better results the multivariate polyomial iterpolatio. I dimesio, Berrut ad Mittelma [] suggested that it might be possible to avoid poles by usig ratioal fuctios of higher degree. They cosidered algorithms which fit ratioal fuctios whose umerator ad deomiator degrees ca both be as high as ay positive iteger. As observed i Berrut ad Mittelma [], every such iterpolat ca be give i the barycetric form r(x) = w i x x i f (x i ) for some real values w i. Thus it is eough for good approximatio rates to fid the weights w 0,w,...,w to specify the fuctio r. There was aother suggestio by Berrut [3], simply to take givig w i x x i w i = ( ) i, k = 0,,..., r(x) = ( ) i f (x i ) x x i Berrut showed that () has o poles i R. See also Berrut [4],[5]. ( ) i x x i. () Floater ad Horma [] reported that there is a whole family of barycetric ratioal iterpolats with arbitrarily high approximatio orders, icludig the iterpolat () as a special case. The costructio is as follows. Choose ay iteger Correspodig author osmarasit@mu.edu.tr c 05 BISKA Bilisim Techology

2 0 O.R.Isik, Z Guey ad M. Sezer : A multivariate ratioal iterpolatio with o poles i R m d with 0 d, ad for each i = 0,,..., d, let p i deote the uique polyomial of degree at most d that iterpolates f at the d + poits x i,x i+,...,x i+d. The let r(x) = d λ i (x)p i (x) d λ i (x) () λ i (x) = ( ) i (x x i )...(x x i+d ). For each d = 0,,,...,, oe of () has ay poles i R. I additio, for fixed d the iterpolat has approximatio order O(h d+ ) as h 0, h := max (x i+ x i ) (3) 0 i as log as f C d+ [a,b]. Floater ad Horma [] used the followig costructio to show () that has o poles i R. O multiplyig the umerator ad deomiator i () by the product ( ) d (x x 0 )...(x x ), we obtai which we ca also express i the form r(x) = d µ i (x)p i (x) d µ i (x) µ i (x) = ( ) d (x x 0 )...(x x )λ i (x), (4) µ i (x) = i =0 (x x ) k=i+d+ (x k x). (5) As usual, a empty product i (5) has value. After tha, Floater ad Horma [] aalyzed the covergece ad the results are give i the followig theorems. Theorem. For all d, 0 d, the ratioal fuctio r i (4) has o poles i R. Theorem. Suppose d ad f C d+ [a,b], ad let r be the ratioal fuctio i () ad h be as i (3). If d is odd the f (d+) r f h d+ (b a). (6) d + If d is eve the f (d+) f (d+) r f h d+ (b a) +. (7) d + d + Theorem 3. Suppose d = 0, f C [a,b] ad the local mesh ratio { } β := max mi xi+ x i x i+ x i, i x i x i x i+ x i+ c 05 BISKA Bilisim Techology

3 NTMSCI 3, No., 9-8 (05) / is bouded whe h 0. Let r be the ratioal fuctio i (). If is odd the If is eve the r f h( + β)(b a) f. ( r f h( + β) (b a) f + f ). I this study, as a extesio of the methods of Floater ad Horma [] i the uivariate case, a family of multivariate ratioal iterpolats which have o poles i R m is costructed. The method is give with a priori error estimate uder low regularity assumptios. We ote that the set of iterpolatio odes K form a tesor product grid. Prelimiaries. Iterpolatio Polyomials Let us cosider the + pairs x i i R. The problem is to fid a iterpolatig polyomial such that p m f (x i ) = a 0 + a x i + + a m x m i = y i, i = 0,,...,. The poits x i are called iterpolatio odes. The followig two theorems ca be foud ay umerical aalysis text, (see, e.g., Quarteroi et al. [6]). Theorem 4. Give + distict odes x 0,x,...,x ad + correspodig values y 0,y,...y, the there exists a uique polyomial p f P such that p f (x i ) = y i for i = 0,,...,. I the ext theorem, Lagrage characteristic polyomials are used which are defied as l i P : l i (x) = =0 i (x x ) (x i x ), i = 0,,...,. Theorem 5. Let x ad the abscissas x 0,x,...,x be cotaied i a iterval [a,b] o which f ad its first + derivatives are cotiuous. The there exists ξ x (a,b), which depeds o x, such that f (x) p f (x) = ( + )! =0 (x x ) f (+) (ξ x ). I m dimesios, iterpolatio polyomial of a fuctio f is defied similar to dimesio, see e.g. [7]. Let us cosider the distict poits x i = (x i,...,xm i ) i Rm. Let ϕ,...,ϕ m deote m liearly idepedet fuctios i C(R m ). The iterpolatig problem is to determie a,a,...a m such that for i. a ϕ (x i ) + a ϕ (x i ) + + a m ϕ m (x i ) = f (x i ) The followig paragraph was give by Mößer ad Reif i [8]. The space dimesio m is assumed as fixed ad greater tha. Let K = deote a partitio of the iterval [0,b ]. Give the tesor product partitio K = m K of = { x i 0 x < x < < x b,i =,..., } m = [0,b ]. For each coordiate directio =,,...,m, uivariate iterpolatio operator mappig a fuctio f with a essetially bouded weak th derivative to the uique polyomial p = I f of order, iterpolatig f o K, is deoted by I : W P (8) c 05 BISKA Bilisim Techology

4 O.R.Isik, Z Guey ad M. Sezer : A multivariate ratioal iterpolatio with o poles i R m The, the iterpolatio polyomial ca be writte as p (x) = i= f (x i )l i (x), l,...,l are Lagrage polyomials. The error operator related to I is defied by E := I. The, a upper boud for the error of polyomial iterpolatio ca be writte i the form E f w t f ( ) w = (x x ) (x x t! ad shows the supremum orm o [0,b ]. For m dimesios, let e deote the th uit vector, ad let x = x x e. (8) is exteded to a operator I f (x) = i= ) f (x + x i e )l i (x), x K which acts oly the th compoet of a give multivariate fuctio f ad all other compoets are treated as costats. The the tesor product iterpolatio I := I m I : W (K) P iterpolates f o K ad p := I f =I m I f is uique. Let deote sup-orm o K. The error operator E ca be give as E = ( E m ) αm ( E ) α (9) α= α = (α,α,...,α m ) N m is a multi-idex with maximal compoet α = maxα i. Thus, the upper boud of the error is obtaied as E f α= w α wα vα m m t α t α Let us write α f := α α α m m f ad t α := t t...t m m....t α m m m α α i α m m m f Theorem 6. For f W (K) the tesor product iterpolatio error o the box is bouded by f I f α= w α t α α f, w = (w,w,...,w m ), t = (t,t,...,t m ) ad α = (α,α,...,α m m ). 3 Ratioal iterpolatig fuctio i m dimesios We will seek a approximate ratioal fuctio for the give fuctio f. Let d Z ad 0 d. For m ad 0 i d, let p i,i...,i m be the polyomial which iterpolates f at Let { (x t,x t,...,x t m m ) : i t i + d, m }. r = d i =0 d i =0 d i =0 d m λ i,i,...,i m p i,i,...,i m i m =0 d i =0, (0) d m λ i,i,...,i m i m =0 c 05 BISKA Bilisim Techology

5 NTMSCI 3, No., 9-8 (05) / 3 λ i,i,...,i m (x) = i +d ( x x i=i m i k ( ) k= )... i m+d m ( i=i m x m x m We will show that the fuctio r i (0) is defied i R m. First, let us defie a ew fuctio µ i,...,i m as µ i,...,i m (x) = i ( x x k ) ( ) x k x k=0 k=i +d + ( im xm x k ) ( ) m x k m x m. k=0 k=i m +d m + k=0 ) () If the idex set is empty, we will agai uderstad its product has the value. O multiplyig both umerator ad deomiator by ( ) m m d k ( ) k=0 x x k m ( ) x m xm k we obtai The deomiator of () is equal to r = d i =0 d i =0 d i =0 d i =0 k=0 d m µ i,i,...,i m p i,i,...,i m i m =0 d i =0. () d m µ i,i,...,i m i m =0 d m µ i (x ) i m =0 µ i (x m ). We see from Theorem that each compoet of the deomiator is greater tha zero, ad the followig results hold. Theorem 7. For all d,d,...,d m, 0 d,d,...,d m, the ratioal fuctio r i () has o poles i m [a i,b i ]. i= Corollary. For all d,d,...,d m, 0 d,d,...,d m, the ratioal fuctio r i () has o poles i R m. Let us cosider the ratioal iterpolatio fuctio r. Let the iterpolatio polyomials i (0) are all be tesor product iterpolats. First, let us discuss the covergece order for dimesios. Let d,d > 0. Let I, I ad I 3 be defied as I,k = {i : i α d k,0 i d k } I,k = {i : α d k + i α,0 i d k }, I 3,k = {i : α + i,0 i d k }, k =, as i []. Let h = max x i+ x i ad h = max y i+ y i. Sice d,d > 0, the followig result ca be obtaied 0 i 0 i by []: d d λ i, (x,y) =0 = s(x,y) x x i y y i (3) µ (x)s(y) s(y), I, or d d =0 λ i, (x,y) = x x i y y i s(x,y) x x i y y i µ (y)s(x) x x i y y i d!h d + y y i s(x) d!h d + x x i, I,. (4) c 05 BISKA Bilisim Techology

6 4 O.R.Isik, Z Guey ad M. Sezer : A multivariate ratioal iterpolatio with o poles i R m Usig the defiitio of µ yields ad µ i (x) s(x) µ i (y) s(y) h t (5) h t (6) t = mi x i+ x i ad t = mi y i+ y i. To see this, first let x (x α,x α+ ) ad i I,. The, µ i (x) 0 i 0 i µ α d (x). Sice x xα d h x x α+d +, t we get the desired result. A similar argumet ca be made for I 3,. We ow deduce the followig theorem for the dimesioal case. Theorem 8. Suppose that d,d are positive ad f W + ( [0,b i ]). Let the iterpolatio odes be { i= (xi,y ) : 0 i, }. The, f r ( d )( d )h d + d+ f t (d + ) x d + + ( d )( d )h d + d+ f t (d + ) y d + + ( d )( d )h d + h d + d +d + f (d + )(d + ) x d+ y d + h = max x i+ x i,h = max y i+ y i, 0 i 0 i t = mi x i+ x i, t = mi y i+ y i. 0 i 0 i Proof. Sice the error fuctio f r is zero o the iterpolatio poits, it is eough to fid the error o the set S := { (x,y) : (x,y) [a,b ] [a,b ]\ { (x i,y ) : 0 i, }}. The fuctio λ i,i i () is well-defied o S ad we ca write the error fuctio as f (x,y) r(x,y) = d d i =0 i =0 λ i,i (x,y)[ f (x,y) p i,i (x,y)] d d i =0 i =0 λ i,i (x,y). (7) We will boud the error fuctio by fidig a upper boud o the umerator ad a lower boud o the deomiator of this quotiet. The fuctio E = f p i,i f ca be writte by (9) as E = E + E E E. Thus, the umerator of (7) is bouded by ( f r)(x,y) + (d + )! (d + )! d i =0 d i =0 d i =0 d i =0 w i (y) λ i,i (x,y) d + f x d + d d i =0 i =0 w i d + f y (x) d + d d λ i,i (x,y) i =0 i =0 d d d +d + f i =0 i =0 x d + y d + (d + + )!(d + )! d i =0 d i =0 λ i,i (x,y). c 05 BISKA Bilisim Techology

7 NTMSCI 3, No., 9-8 (05) / 5 We ow obtai the iequality by applyig (3-6) ( f r)(x,y) + + (d + )! (d + )! d i =0 d i =0 d!h d + d i =0 s(y) d i =0 d!h d + w i (y) y y i s(x) (d + )!(d + )! hd + d + w i (x) x x i d i =0 d i =0 d!h d + d!h d + d + f x d + d + f y d + d +d + f x d + y d + d d µ i (y) d + f i =0 i =0 s(y) x d + + hd + d d µ i (x) d + f d + i =0 i =0 s(x) y d + + hd + h d + d d d +d + f (d + )(d + ) i =0 i =0 x d + y d + h d + ( d )( d ) d + f t (d + ) x d + h d + + ( d )( d ) d + f t (d + ) y d + + ( d )( d )h d + h d + d +d + f. (d + )(d + ) x d + y d + Simplifyig the above iequality by usig (4) yields the desired result. ( ) m Theorem 9. Suppose that d,d,...,d m $ are all positive ad f W + [0,b i ]. The f r m ( d i ) α = ( h t ) α h α(d+) α(d+) f, α(d + ) = (α (d + ),α (d + ),...,α m (d m + )), h = (h,h,...,h m ),t = (t,t,...,t m ). Proof. It is proved by usig the similar steps as i Theorem 8 ad the iequalities µ i,i,...,i m (x) s(x), 0 i d, 0 m, d s(x) = d µ i (x ) d µ i (x ) µ i (x m ). 4 Numerical Examples I this sectio, several umerical examples are give to illustrate the properties ad effectiveess of the method. We compare the approximate solutio with polyomial iterpolatio ad piecewise polyomial iterpolatio. All calculatios were made usig Maple 9. c 05 BISKA Bilisim Techology

8 6 O.R.Isik, Z Guey ad M. Sezer : A multivariate ratioal iterpolatio with o poles i R m Fig. : Approximatio error for f (x,y) = 3(x + y) / for = 6 ad d = d = 4. Fig. : Approximatio error for f (x,y) = 3(x + y) / for = 0 ad d = d = 7. Example. We wat to approximate f (x,y) = 3(x+y) o [0,] [0,] which was give as a example by Mößer ad Reif [8]. Selectig the odes {(x i,y ) : x i = i6, y = 6 }, 0 i, 6, the absolute error for = 6 ad d = d = 4 is foud as f r Similarly, takig the equidistat odes, the absolute error for = 0 ad d = d = 7 is obtaied as f r 0 5. The absolute errors for = 6, d = d = 4 ad = 0, d = d = 7 are plotted i Figure ad Figure, respectively. Also, the ifiity orms of error fuctio for = 0 ad various d,d are give i Table. Table : The upper bouds of absolute errors for = 0 ad various d,d d = d = 3 d = d = 4 d = d = 5 d = d = E 4 4.4E 5 ) Example. Let us cosider the fuctio f (x,y) = x y ( e (x +y ) o [0,] [0,] dealt with i Mößer& Reif [8]. Let us fid the approximate solutio for = 8, d = d = 5 o equidistat odes. After fidig the approximate fuctio r i (4), the upper boud of absolute error is foud as The error fuctio is plotted i Figure 3. The followig example was give i [9]. f r c 05 BISKA Bilisim Techology

9 NTMSCI 3, No., 9-8 (05) / 7 Fig. 3: Error fuctio for ) f (x,y) = x y ( e (x +y ) o equidistat odes. Fig. 4: Error fuctio for f (x,y) = 3 (9x ) 4e 4 (9y ) (9x ) 4e 49 (9y ) 0 + (9x 7) e 4 (9y 3) 4 5 e (9x 4) (9y 7). We cosider the fuctio f (x,y) = 3 (9x ) e 4 (9y ) (9x ) e 49 (9y ) (9x 7) e 4 (9y 3) 4 5 e (9x 4) (9y 7) o [0,] [0,]. The upper boud of absolute error for = 0, d = d = 6 is obtaied as ad the error fuctio is plotted i Figure 4. Example 3. We apply the method to f r 0.035, f (x,y) = e x cosy, (x,y) [0,] [0,] which we sampled at the equidistat spaced poits. The errors are give below for = 5, d = d = 4ad = 0, d = d = 7 : f r ad f r As a last example, to compare the approximate solutio with piecewise polyomial iterpolatio, we give a example from [6]. Example 4. We compare the covergece of the piecewise polyomial iterpolatio of degree ad ratioal approximatio for = 4, d = d = ad = 8, d = d =, o the fuctio f (x,y) = e (x +y ) o [0,] [0,]. While the errors, for = 4, d = d = with respect to piecewise polyomial iterpolatio ad with respect to the preset method are f r c 05 BISKA Bilisim Techology

10 8 O.R.Isik, Z Guey ad M. Sezer : A multivariate ratioal iterpolatio with o poles i R m ad f r , respectively, the errors, = 8, d = d =, with respect to piecewise polyomial iterpolatio ad with respect to the preset method are f r ad respectively. f r , 5 Coclusio Give ay multivariate ( fuctio f ), oe ca approximate by (4) easily o a box ad estimate its error by Theorem 9 m provided that f W + [0,b i ]. Sice the approximate solutio (4) depeds o polyomial iterpolatio, it may ot coverge. As see from the examples, the method gives good approximatio. More accurate results ca be obtaied for small d i ad. If the fuctio f is a polyomial, the method gives the exactly f sice its iterpolatio polyomial is agai itself. The method is applicable to all multivariate fuctios ad it depeds o m ( d i ) fuctio evaluates. Refereces [] M.S. Floater, K. Horma, Barycetric ratioal iterpolatio with o poles ad high rates of approximatio, Numerische Mathematik, 07 (006) [] J. P. Berrut, H. D. Mittelma, Lebesgue costat miimizig liear ratioal iterpolatio of cotiuous fuctios over the iterval, Comput. Math. Appl., 33 (997) [3] J. P. Berrut, Ratioal fuctios for guarateed ad experimetally well-coditioed global iterpolatio, Comput. Math. Appl., 5 (988) -6. [4] J. P. Berrut, Barycetric Lagrage iterpolatio, SIAM Rev., 46 (004) [5] J. P. Berrut, R. Baltesperger, H. D. Mittelma, Recet developmets i barycetric ratioal iterpolatio. Treds ad Applicatios i Costructive Approximatio(M. G. de Brui, D. H. Mache, ad J. Szabados, eds.), Iteratioal Series of Numerical Mathematics, 5 (005) 7-5. [6] A. Quarteroi, R. Sacco, F. Saleri, Numerical mathematics, Spriger, New York, 007. [7] G. M. Phillips, Iterpolatio ad Approximatio by Polyomials,Spriger, New York, 003. [8] B. Mößer, U. Reif, Error bouds for polyomial tesor product iterpolatio, Computig, 86 (009) [9] A. Sommariva, M. Viaello, R. Zaovello, Adaptive bivariate Chebyshev approximatio, Numer. Algorithms 38 (005) c 05 BISKA Bilisim Techology